I know this might be an avant garde suggestion, but have you tried actually talking to her about this?
If you don't understand what she is struggling with, then you don't know what the situation is. Getting the kids ready for school has a definite time limit on it in which to do everything - it might be completely irrelevant that she has the rest of the day off if she is struggling to get everything necessary done before the kids have to leave the house.
If someone tells you that you have to run a marathon every day, then it's fine if you have all day to do it. But if you have to have finished within a very specific time frame, then it simply might not be possible.
If that's the case, then yes, you are being unreasonable to expect to get a nice relaxing morning whilst she kills herself trying to desperately get everything ready.
If she is dealing with the children if they wake up in the night, or have nightmares, or other night wakings (so that you can get a decent night's sleep before work), then it's natural she'd be shattered in the mornings, and it would be reasonable to expect you to help out.
There are reasons why she might be totally reasonable in her request, and there might be reasons why she is being unreasonable.
But you won't know until you actually have an adult, non competitive, non aggressive discussion with her about it.
What isn't helpful is to point blank reject the idea that you should help with your children before you go to work, purely on the basis that you have work afterwards.
If the children are in school now, perhaps you could both sit down and discuss whether it's worth her returning to work, or looking for something part time. In which case you would both need to deal with all childcare issues 50/50 (or whatever split depending on whether she goes back ft or pt).
If that isn't possible because you need someone to do child pick ups, drop offs , someone to look after the kids when they're sick, then you need to re-evaluate your view on what your wife actually does. It isn't just that she is sat at home on her arse all day whilst you...sit at work behind a desk all day (? lol) - what she is actually doing is making herself available at any time in case a child care issue arises. And that in turn allows you to work undisturbed, and without having to pay for child care, or arrange drop offs, or pay for extra activities to keep the kids busy until you two finish work.